Speaking last month, Chargers head coach Anthony Lynn said Kaepernick fits the team's style of play and, while the franchise are happy with their options of Justin Herbert, Tyrod Taylor and Easton Stick, the former San Francisco 49er is on a workout list.
Kaepernick has been out of the league since the end of the 2016 season, during which he opted to kneel for the national anthem in protest against racial injustice and police brutality.
He was involved in a workout for teams in November last year and his cause received renewed attention amid nationwide protests after the death of George Floyd in police custody on May 25.
Telesco, speaking on a Zoom call with reporters, said the Chargers have no need to look for a new quarterback – as it stands right now.
"No update, we're very comfortable with the quarterbacks we have right now," Telesco said.
"If something changes at some point down the road, yeah, we would look at him and treat him no differently than any other free agent quarterback that's available.
"But workouts for us are needs based, the need is not there right now. As you can see, this is a constantly changing world right now, so you always leave everything open.
"Right now, we're happy with the three guys we have."
Telesco added he is confident about the NFL season going ahead amid the coronavirus pandemic despite complications other team sports in the United States have faced.
"The protocols we're following, the distancing, I felt really good. People holding each other accountable, it's a lot of change for us but we felt really good about it," he told the media.
"Then you wake up in the morning and I saw that the Miami Marlins had a lot of players test positive and a couple of coaches.
"The head trainer for the Minnesota Vikings, he tested positive, the Rockets CEO, I know how well they do things there. We're all under the same protocol. That's reality.
"The reality hits hard that you just cannot let your guard down at any point, any time, it's going to be 24/7 that we're fighting this.
"I will say that Anthony and I, we feel a real responsibility to keep everybody here safe. Players, coaches and a very large support staff of people.
"What we've done so far, all the planning that's gone into this so far for us has been very efficient and we're going to keep working through it every day."